I think it is zero (i.e. its fractal dimension is less than 3)... however I don't have proof and it seems to get denser as you get towards the centre. So there might be a depth at which no points can escape.

Here is some kind of experiment to support Tglad's answer. When maximum number of iterations is very high then Mandelbox starts to be empty. Theoretically when maxiter will be infinite number, then Mandelbox will disappear.

My thought on this is that having zero volume (or D<3) does not necessarily mean that the object should be invisible. A plane has zero volume and is visible, and any 'shell' fractal is still visible:and 'foam' fractals are still visible:

For fractals that definitely are ultimately invisible (their 2D screen projection has fractal dimension < 2), such as trees, an interesting topic is that there are different ways to render it that keep it visible without it being an approximation http://tglad.blogspot.com.au/2012/12/dimension-aware-rasterising.html (I even made a paper on it 'Improving image clarity using local feature dimension'), but it has never been attempted in 3D.

You are right about disappearing. The fractal will be still visible but all remaining surfaces will have zero thickness. That's why they disappear when is used used raymarching with bisection algorithm.

Program FractaloScop can get to 0.0000001.If we imagine this structural resolution as a 1 cm, 1.0 represents 100km, 2PI's MandelBulb is virtualy a 628km wide ball.If we imagine it instead as a 0.1 mm, 1.0 still represents 1km and MandelBulb shrinks to a 6.28km ball.And so on...

The same logic can be applied to MandelBox too, but there is a catch.MandelBox can be easily resized and so heavily vary in size.I have changed almost each line of equation (just to get more readable inner space) and the default size for my program is now set to 12.So the mandelbox for me is a virtual box sized 1200km when the smallest geometry detail rendered as a pixelized block is a 1cm.